Audience members listen to Democratic presidential candidate Hillary Rodham Clinton as she speaks before the National Urban League, on Friday in Fort Lauderdale, Fla. (Wilfredo Lee/AP)

Following President Obama’s reelection in 2012 — a win powered in part by historic black voter turnout — Republicans launched an aggressive outreach effort aimed at African American voters. Never again, they vowed, would they concede that sort of overwhelming demographic advantage to Democrats.

Now, heading into an election without Obama at the top of the ticket, many Republicans say they are positioned to make inroads with black voters.

That sense of opportunity was clear Friday when the party establishment’s favorite, former Florida governor Jeb Bush, addressed the annual meeting of the National Urban League in his home state. He spoke to the predominantly African American crowd about his record of helping blacks and his plans to do more. He touched on his removal of the Confederate flag from the state capitol grounds, his gubernatorial record on diversity and how his education reforms helped black students.

“If we don’t create an education system that allows young people to reach [the rungs on the ladder of success], we’re setting them up for a lifetime of failure,” he said. “So you and I have to call this situation what it is: the worst inequality in America today, and the source of so many other inequalities.”

Bush, and his party, think that sort of pitch may resonate with black voters. Some Democrats worry they may be right.

Speaking before Bush on the same stage, Democratic front-runner Hillary Rodham Clinton used her address Friday to draw a sharp contrast with him, invoking the Black Lives Matter movement. Without naming her GOP rival, Clinton cast him as a poor choice for black voters on health care, entitlements and voting rights.

“You cannot seriously talk about the ‘right to rise,’ ” she said, invoking a Bush campaign theme and the name of a super PAC founded by his allies, “and support laws that deny the right to vote.” A Bush spokesman later said that Clinton took a “false cheap shot.” But her audience broke into hearty applause.

It was a morning that showcased the challenges Republicans face in pursuing black voters — and their determination to claim what they say is an increasingly strong opportunity to win over African American support.

“We’re at a moment in time when the black community is receptive,” said former Republican National Committee chairman Michael Steele, the first black man to hold that job. “As we transition from the Obama administration and the Obama leadership, they are looking. They have not sold themselves on Hillary. They have not bought into the Bernie Sanders socialist view of the world. They are suspicious of Martin O’Malley.”

It would not take many black votes to complicate the Democrats’ electoral map. Neil Newhouse, a Republican pollster working for a super PAC supporting Bush, said in an e-mail that the “demographic challenges” facing the eventual GOP nominee are “real and significant,” but fixable.

“The payoff can be significant,” Newhouse said. “It doesn’t take much of a swing in minority votes to make a difference. Winning even 10 to 14 percent of African American votes in states like Ohio, Florida or Virginia could put those states in the GOP column in ’16.”

Democrats have left few openings for the GOP on voting rights. One of the major drivers behind the 2012 black turnout in Ohio, a key swing state, was the decision by Republicans to cut back on early-voting days.

GOP nominee Mitt Romney won just 17 percent of the nonwhite vote in 2012, down a bit from the 19 percent John McCain won in 2008 and a steeper drop from the 26 percent George W. Bush won in 2004. Had Romney performed as well as Bush among nonwhite voters, he, too, would have won Ohio.

Jeb Bush, the 43rd president’s brother, has not shied from questions about race, confronting the issue more directly than many past GOP candidates have. He has talked about Sandra Bland, whose arrest and death in a Texas jail cell in July prompted an outcry; he has opined on the Black Lives Matter movement; and he has recounted in Spanish the taunting his half-Hispanic children endured because of their skin color.

Cuyahoga County and Cleveland, which will host the first GOP primary debate next week, have been laboratories for Republicans’ theory that they can turn the tide. Obama won the Ohio county in 2012 by 256,613 votes, boosting him to a 166,272-vote statewide win. In some Cleveland precincts, Romney won no votes at all.

Just two years later, Republican Gov. John Kasich aggressively pitched black voters, the local party started a conversation with some open-door debates — and Kasich won the county by 22,333 ballots, a reelection landslide that has become part of his story to GOP primary voters.

Some Republicans also see Marco Rubio, a youthful Cuban American senator from Florida, as a candidate who can build new bridges to black voters. After a campaign stop in Greenville, S.C., this week, Rubio said he thinks the country has a “painful, complicated” history with racism.

“I think its impacts are still felt in many communities across the country,” Rubio told reporters. He acknowledged that some black communities in South Florida feel “deeply disadvantaged,” and with reason.

But having the right messenger is only part of the equation for improving the party’s standing with black voters, Republicans say. To that end, some have embarked on changing policies.

Kasich responded to the fatal police shooting of Tamir Rice, a 12-year-old Cleveland boy, by convening a task force to review law enforcement conduct. Sen. Tim Scott (S.C.), the upper house’s only black Republican, said in an interview that his party is leading on criminal justice reform and that black voters are taking notice.

“Republican governors have done a magnificent job on that,” Scott said. “I’ve got legislation on requiring body cameras for police officers, which resonates very well in the African American community. I hope we are building new connections, because the effects of Democratic policies on black communities have frankly not been successful.”

But even as Republicans map this new territory, the ground may be shifting. Bush learned that recently when he balked at how Black Lives Matter activists had jeered the term “all lives matter” at a progressive conference. He didn’t mention the phrase in his Friday speech.

Asking for the political conversation to center on white privilege, as those activists demanded, was too much, even for some prominent black Republicans. “I certainly think that black lives matter,” GOP presidential candidate Ben Carson, the race’s only African American contender, said in an interview. “Of course, all lives matter. The sooner we can get away from the politics of identity in everything, the better off we are.”

Carson and Bush, the only Republican presidential candidates to address the National Urban League on Friday, did not mention the Black Lives Matter movement in their remarks. Clinton, along with her Democratic opponents O’Malley and Sanders, did.

To shrink the Democrats’ 10-to-1 or 20-to-1 margins among black voters, Republicans know they will have to do plenty of the sort of explaining Carson offered. They say they are ready — and so is a post-Obama black electorate.

“We have a saying on my radio show: ‘Save the saveable,’ ” said Herman Cain, a black businessman who ran for the 2012 Republican nomination and briefly led the race. “You’re not going to save everybody with the facts and the truth. But you can save a whole lot of them.”

Weigel and Lowery reported from Washington. Ed O’Keefe contributed to this report.